Thousands if not tens of thousands, or more, of points across the world become in essence, hyper-spacial tesseracts where time become a fluid thing and it is possible to slip forwards and backwards in history to various points potentially going back as far as the Snowball Earth of the Cryogenian era 650 million years ago.

The way it works is only living things can traverse these rifts and more often than not with some primitive sense of guidance. If one was in the middle of the street and you wanted to cross it, you could do so safely if you had your own time destination in mind, but if you were thinking that you could really use some surf and sand, you could find yourself standing on the shore of some grand Silurian ocean, or any when if let your mind wander. The only limit is you will never find yourself immediatly lethal as your instincts will direct you away from such dangers.

Of course you can try and focus on another era or time and you'll end up as close to it as time will allow, and the way to get back home is simply to think of your own era and travel back through the tesseract.

There's really no set timetable and the instant of the event every world became its own, all mingled together in a vast and traversable web of time.

Not every world leads to every other world, but you can always find your way home, and as for travel to recent times... well... not saying its not an option, but the closest you can get to our own history at least via our own Earth is sometime around the middle of the last ice age... which mind you isn't saying its impossible just we're not directly connected to another "Modern Earth".

Any living thing can traverse it, though animals can do so with the greatest ease and some level of thought, even if basic, is needed to guidance. Also, all living things will instinctively know where one is, if only by their subconscious knowing that there is a something that goes somewhere round about.

Size varies and they can vanish and appear almost at random (many will be mostly static though).

So, how does this fucked up situation shape things in the short term, midterm, and long term...

This is a chance to be creative. Give it your all!

Though I will make it clear... time moves at the same rate and once the web "formed" each Earth became an independent timeline.

And lets leave hand-waving any super plagues as an option. "Everything dies of disease" isn't very gripping. Or it can be... up to you.